84 f(-p1) + f(-p2) Z(b(p3) + b(p4)) + Z(3 * (ν(p5) + ν(p6)))

84.1 Di-boson production, processes 61–89

These processes represent the production of a diboson pair V 1V 2, where V 1 and V 2 may be either a W or Z∕γ*. All the processes in this section may be calculated at NLO with the exception of nproc=69. There are various possibilities for the subsequent decay of the bosons, as specified in the sections below. Amplitudes for the V 1V 2 process are taken from ref. [1]. We also include singly resonant diagrams at NLO for all processes in the case zerowidth = .false.. For more details on this calculation, please see Refs. [23].

For processes 62, 63, 64, 65, 74 and 75 the default behaviour is that the hadronic decay products of the bosons are clustered into jets using the supplied jet algorithm parameters, but no cut is applied on the number of jets. This behaviour can be altered by changing the value of the variable notag in the file src/User/setnotag.f.

Calculations of processes 61, 71, 76, 81 and 82 can be performed at NLO by subtraction, zero-jettiness slicing and qT-slicing. They can be computed at NNLO using zero-jettiness slicing and qT-slicing, as described in ’Non-local slicing approaches for NNLO QCD in MCFM’,[4]. For processes 61, 81 and 86 the NNLO corrections include glue-glue initiated box diagrams which first contribute at order αs2.

84.1.1 ZZ production, processes 81–84, 90

The Z’s can either both decay leptonically (nproc=81), one can decay leptonically while the other decays into neutrinos (nproc=82) or bottom quarks (nproc=83), or one decays into neutrinos and the other into a bottom quark pair (nproc=84). In process 83 the mass of the b-quark is neglected. Note that, in processes 8384, the NLO corrections do not include radiation from the bottom quarks that are produced by the Z decay. In process 90 the two Z bosons decay to identical charged leptons, and interference effects between the decay products of the two Z bosons are included. In all cases these processes also include the contribution from a virtual photon.

When removebr is true in process 81, neither of the Z bosons decays.

This process has been treated in several papers, [2354]. Processes 81 and 82 can be calculated at NNLO. The NNLO calculation includes contributions from the process gg ZZ that proceeds through quark loops. The calculation of loops containing the third quark generation includes the effect of both the top and the bottom quark mass (mt,mb0), while the first two generations are considered massless. For numerical stability, a small cut on the transverse momentum of the Z bosons is applied: pT(Z) > 0.1 GeV. This typically removes less than 0.1% of the total cross section. The values of these cutoffs can be changed by editing src/ZZ/getggZZamps.f and recompiling.

84.1.2 Anomalous WWZ

It is possible to specify anomalous trilinear couplings for the W+W-Z and W+W-γ vertices that are relevant for WW and WZ production. To run in this mode, one must set zerowidth equal to .true. and modify the appropriate lines for the couplings in input.ini (see below). Note that, at present, the effect of anomalous couplings is not included in the gluon-gluon initiated contributions to the WW process.

The anomalous couplings appear in the Lagrangian, L = LSM + Lanom as follows (where LSM represents the usual Standard Model Lagrangian and Lanom is taken from Ref. [6]): PIC

where Xμν μXν - νXμ and the overall coupling factors are gWWγ = -e, gWWZ = -ecotθw. This is the most general Lagrangian that conserves C and P separately and electromagnetic gauge invariance requires that there is no equivalent of the Δg1Z term for the photon coupling.

In order to avoid a violation of unitarity, these couplings are often included only after suppression by dipole form factors,

PIC

where ŝ is the vector boson pair invariant mass and Λ is an additional parameter giving the scale of new physics, which should be in the TeV range. These form factors should be produced by the new physics associated with the anomalous couplings and this choice is somewhat arbitrary. The use of the form factors can be disabled as described below. The file input.ini contains the values of the 6 parameters which specify the anomalous couplings. If the input file contains a negative value for the form-factor scale then the suppression factors described above are not applied.

84.2 Plotter

nplotter_ZZlept.f is the default plotting routine.

84.3 Example input and output file(s)

input84.ini process84.out

References

[1]    L.J. Dixon, Z. Kunszt and A. Signer, Helicity amplitudes for O(αs) production of W+W-, WZ, ZZ, Wγ, or pairs at hadron colliders, Nucl. Phys. B531 (1998) 3 [hep-ph/9803250].

[2]    J.M. Campbell and R.K. Ellis, An Update on vector boson pair production at hadron colliders, Phys. Rev. D 60 (1999) 113006 [hep-ph/9905386].

[3]    J.M. Campbell, R.K. Ellis and C. Williams, Vector boson pair production at the LHC, JHEP 07 (2011) 018 [1105.0020].

[4]    J.M. Campbell, R.K. Ellis and S. Seth, Non-local slicing approaches for NNLO QCD in MCFM, 2202.07738.

[5]    R. Boughezal, J.M. Campbell, R.K. Ellis, C. Focke, W. Giele, X. Liu et al., Color singlet production at NNLO in MCFM, Eur. Phys. J. C 77 (2017) 7 [1605.08011].

[6]    L.J. Dixon, Z. Kunszt and A. Signer, Vector boson pair production in hadronic collisions at order αs : Lepton correlations and anomalous couplings, Phys. Rev. D60 (1999) 114037 [hep-ph/9907305].